MATHEMATICAL THINKING
1. Mathematical Attitude
a. Attempting to grasp one’s own problems or objectives or substance clearly, by oneself
· Attempting to have questions
· Attempting to maintain a problem consciousness
· Attempting to discover mathematical problems in phenomena
b. Attempting to take logical actions
Assume that children have explained that in the case of a triangle such as the one in the figure to the right (1), area = base side × height ÷ 2.
a. Attempting to express matters clearly and succinctly
· Attempting to record and communicate problems and results clearly and succinctly
· Attempting to sort and organize objects when expressing them
b. Attempting to seek better things
· Attempting to raise thinking from the concrete level to the abstract level
· Attempting to evaluate thinking both objectively and subjectively, and to refine thinking
· Attempting to economize thought and effort
1. Mathematical Method
a. Inductive Thinking
Meaning
Inductive thinking is a method of thinking that proceeds as shown attempting to gather a certain amount of data, working to discover rules or properties in common between these data,inferring that the set that includes that data (the entire domain of variables) is comprised of the discovered rules and properties and confirm the correctness of the inferred generality with new data
Example : (Mathematics for Junior High School 1 page 53)
Converting a mixed to improper fraction.
We can use the following formula to convert a mixed fractions to improper fraction.
The function of this formula is to find a more simple way to solve the problem.
a. Analogical Thinking
Meaning
Analogical thinking is an extremely important method of thinking for establishing perspectives and discovering solutions.
What is Analogical Thinking?
Given Proposition A, one wants to know its properties, rules or solution methods.
However, when one does not know these things, one can recall an already known Proposition A’, which resembles A (assuming that regarding A’, one already knows the properties, rules, solution method, and so on, which are referred to as P’). One then works to consider what can be said about P’ of A’, and with respect to A as well.
Example : (Mathematics for Junior High School 1 page 55)
So, from that example we can conclude that to simplyfying the fraction is by dividing the numerator and denominator by GCD (the greatest common divisor) of a and b.
a. Deductive thinking
This method of thinking uses what is already known as a basis and attempts to explain the correctness of a proposition in order to assert that something can always be stated.
b. Integrative Thinking
Rather than leaving a large number of propositions disconnected and separate, this thinking method abstracts their essential commonality from a wider viewpoint, thereby summarizing the propositions as the same thing.
Example : (Mathematics for Junior High School 1 page 117)
A book costs Rp 3,000.00. Ningsih’s money is worth only to buy 10 books. If the book prices decrease into Rp 2,500.00, how many books she can buy?
If one book prices is h, and Ningsih’s money only enough to bought 10 books, we can write that Ningsih’s money is 10 x h .
If the book prices decrease into Rp 2,500.00, the number of books that Ningsih can buy is,
a. Developmental Thinking
Developmental thinking is when one achieves one thing, and then seeks an even better method, or attempts to discover a more general or newer thing based on the first thing. There are two types of developmental thinking. Type I Developmental Thinking: Changing the conditions of the problem in a broad sense. By “changing the conditions of the problem,” this means:
· Change some conditions to something else, or try loosening the conditions.
· Change the situation of the problem. Type II Developmental Thinking: Changing the perspective of thinking
Example : (Mathematics for Junior High School 1 page 55)
The side of a square is 16 cm. Then, each angle is torn offf 2 cm. Determine the perimeter and the area of the torn off square.
Solution :
The area of square before it is torn off is 16 cm x 16 cm = 256 cm2 .
The area of each torn off part is 2 cm x 2 cm = 4 cm2.
Therefore, the total area of torn off parts = 4 x 4 cm2 = 16 cm2.
The area of figure after it is torn off = 256 -16 = 240 cm2
The perimeter of the figure after it is torn off is :
K = 4 x (2+12+2)= 4 x 16= 64 cm
a. Abstract thinking (thinking that abstracts, concretizes, idealizes, and thinking that clarifies conditions)
Abstract thinking is a method of thinking that, first of all, attempts to elicit the common properties of a number of different things.
Example : (Mathematics for Junior High School 1 page 13)
Addition properties on integers.
· Commutative law
If a and b are the integers, then a+b=b+a
Example : 5+(-7)=(-7)+5=-2
· Assosiative law
If a, b and c , are the integers, then (a+b)+c=a+(b+c)
Example : [7+(-2)]+8=7+[(-2)+8]=13
· Identity element
In the integer, there is an identity element 0 such that a+0=0+a=a
Example : 8+0=0+8=8
b. Thinking that Simplifies
Thinking that Simplifies 1:
Although there are several conditions, and although one knows what these conditions are, when it is necessary to consider all of the conditions at once, sometimes it is difficult to do this from the start. In cases like this, it is sometimes beneficial to temporarily ignore some of the conditions, and to reconsider the problem from a simpler, more basic level. This type of thinking is referred to as “thinking that simplifies.”
Thinking that Simplifies 2:
Thinking that replaces some of the conditions with simpler conditions, is also a type of thinking that simplifies. Keep in mind, however, that general applicability must not be forgotten during the process of simplification. Although the problem is simplified, there is no point in simplifying to the extent that the essential conditions of the original problem or generality are lost. This applies to idealization as well.
Example : (Mathematics for Junior High School 1 page 123)
Fandi bought a USB flash disk for Rp 370,000.00. if he got 15% discount, how much was the discount? How much is the sale price of the USB after getting a discount?
Solution :
Fandi got 15% discount for Rp 370,000.00.
Thus, 15/100 x Rp 370,000.00=Rp 55,000.00.
Therefore, the sale price of the USB is 370,000-55,000=Rp314,500.00
The simple way to find the price is
100%-15%=85%
85% x 370,000= Rp314,500.00
c. Thinking that generalizes
This type of thinking attempts to extend the denotation (the applicable scope of meaning) of a concept. This type of thinking also seeks to discover general properties during problem solving, as well as the generality of a problem’s solution (the solving method) for an entire set of problems that includes this problem.
Example : (Mathematics for Junior High School 1 page 115)
a. Thinking that specializes
Thinking that specializes is a method that is related to thinking that generalizes, and is the reverse of generalization.
In order to consider a set of phenomena, this thinking method considers a smaller subset included in that set, or a single phenomenon in that set. The meaning of specialization is clarified by thinking about when it is used and how it is considered. Thinking that specializes is used in the following cases:
· By changing a variable or other factor of a problem to a special amount without losing the generality of the problem, one can sometimes understand the problem, and make the solution easier to find.
· By considering an extreme case, one can sometimes attain a clue as to the problem’s solution. The result of this clue or method can then be used to assist in finding the general solution.
· Extreme cases or special values can be used to check whether or not a possible solution is correct.
Example : (Mathematics for Junior High School 1 page 77)
Properties of multiplication of the fraction.
Distributive
a x (b+c) = (axb)+(axc), where a, b and c are fractions.
b. Thinking that symbolize
Thinking that symbolize attempts to express problems with symbols and to refer to symbolized objects. This type of thinking also includes the use and reading of mathematical terms to express problems briefly and clearly. This type of thinking proceeds one’s thought based on the formal expression of problems.
Example : (Mathematics for Junior High School 1 page 301)
The perimeter of the square is 4s.
The area of the square is s2
s in here is symbol of side.
c. Thinking that express with numbers, quantifies, and figures
Rather than giving children only numerical expressions, and simply teaching them how to process the numbers, it is necessary for them to start at the stage before quantification, and to have them think about how to quantify the information. What is Thinking that Express with Numbers and Quantifies? This thinking takes qualitative propositions and understands them through qualitative properties. Thinking that selects the appropriate quantity based on the situation or objective is thinking that express with quantities. Thinking that uses numbers to express amounts of quantities is thinking that express with numbers. Conversion to numbers makes it possible to succinctly and clearly express amounts, thereby making them easy to handle. These types of thinking are summarized and referred to as “thinking that express with numbers and quantifies.” In addition to quantification, thinking that expresses problems with figures is also important. What is Thinking that Express with Figures? This thinking replaces numerical propositions and the relationships between them with figures. Situations, propositions, relationships, and so on are replaced with figures and the relationships between them. This type of thinking is referred to as “thinking that expresses with figures.”
Example : (Mathematics for Junior High School 1 page 171)
A bamboo of 15 meters long is cut into two parts. The first part is 3 meters longer that the other.
We can express the longer part is x+3.
1. Mathematical Content
a. Clarifying Sets of Objects for Consideration and Objects Excluded from Sets, and Clarifying Conditions for Inclusion (Idea of Sets)
This is an important aspect of the idea of sets. For instance, when one counts objects, it is not enough to simply count. It is important to first achieve a solid grasp of the scope of objects to be counted. Also, when grasping a concept such as the isosceles triangle, it is important to determine and clearly indicate the scope of objects under consideration (just one printed triangle, a number of triangles created with sticks, or any triangles one can think of with the presented triangles simply offered as examples).
Example : (Mathematics for Junior High School 1 page 279)
In that book explained various triangles classifying by its abgles and side. Such as a right isosceles triangle, an acute isosceles triangle, a right scalene triangle, an acute scalene triangle an an obtuse scalene triangle.
This is an important aspect of the idea of sets. For instance, when one counts objects, it is not enough to simply count. It is important to first achieve a solid grasp of the scope of objects to be counted. Also, when grasping a concept such as the isosceles triangle, it is important to determine and clearly indicate the scope of objects under consideration (just one printed triangle, a number of triangles created with sticks, or any triangles one can think of with the presented triangles simply offered as examples).
Example : (Mathematics for Junior High School 1 page 279)
In that book explained various triangles classifying by its abgles and side. Such as a right isosceles triangle, an acute isosceles triangle, a right scalene triangle, an acute scalene triangle an an obtuse scalene triangle.
b. Focusing on Constituent Elements (Units) and Their Sizes and Relationships (Idea of Units)
Numbers are comprised of units such as 1, 10, 100, 0.1, 0.01, as well as unit fractions such as 1/2 and 1/3, and are expressed in terms of “how many units” there are. Therefore, focusing on these units is a valid way of considering the size of numbers, calculations, and so on. In addition, it goes without saying that amounts are expressed with various units such as cm, m, l, g, and m2, and that tentative unit can be used. Therefore, when one considers measuring the amount of something, it is important to pay attention to the unit. Also, figures are comprised of points (vertices), lines (straight lines, sides, circles, and so on), and surfaces (bases, sides, and so on). For this reason, thinking that focuses on these constituents, unit sizes, numbers and interrelationships, is important.
(Mathematics for Junior High School 1 page 77)
9 x 7/15 = 9 x 7 x 1/15 = 63/15
c. Attempting to Think Based on the Fundamental Principles of Expressions (Idea of Expression)
Whole numbers and decimal fractions are expressed based on the decimal place value notation system. To understand the properties of numbers, or how to calculate using them, one must first fully comprehend the meaning of the expressions of this notation system. The ability to think based on this meaning is indispensable. When it comes to fractions as well, one must be able to see 3/2 as a fraction that means a collection of three halved objects, or the ratio of 3 to 2 (3÷2). It is necessary to consider measurements of amounts based on the definition of expressing measurements with two units, such as when 3 l and 2 dl is written as “3 l 2 dl,” as well as the definition of writing measurements with different units, as in the case where “10,000 m2 is written as 1 ha.” Also, in order to achieve a concrete grasp of the set of numbers, it is necessary to express numbers in a variety of different concrete models, and to take advantage of knowledge about the definition of these expressions. There is a model referred to as the “number line” that is used for expressing numbers. This involves placing an origin point (0) on a straight line, determining the unit size (1), and using this to correspond numbers to points on the line. Use this model based on the definition of this expression. (Array figure) There are many other models in addition to this one. For instance, array or area figures can be used to model a×b=c.
(Mathematics for Junior High School 1 page 91 on exercise number 18)
-4.32 : (8+4) x 3 = -4.32 : (8 x 3 + 4 x 3)= -4.32 : (24 + 12) = -4.32 : 36
= -0.12
d. Clarifying and Extending the Meaning of Things and Operations, and Attempting to Think Based on This (Idea of Operation)
The “things” referred to here are numbers and figures. For instance, what does the number 5 express? How do you clarify the meaning (definition) that determines what a square is? Also, consider numbers and figures based on this meaning. “Operations” refers to formal operations that are used for counting, the four arithmetic operations, congruence, expansion and reduction (similar), the drawing of figures, and so on. These operations are used to calculate with numbers, to think about the relationship between figures and how to draw them in one’s head. When are computation such as addition used? Since the meaning (definition) is precisely determined, the decision of operations naturally follows from the meaning (definition) of computation, along with the methods and properties of computation. Also, the properties and methods of drawing figures, as well as relationships with other figures, are originally clarified based on the meanings (definitions) of those figures. As the scope of discussion expands from whole numbers to decimal fractions and fractions, operations on these numbers also become applicable to a wider scope, and so their meanings must also be extended. Make sure the meanings of things and operations are concrete. It is important to think about properties and methods based on these meanings. These thoughts must follow when one is thinking axiomatically or deductively.
(Mathematics for Junior High School 1 page 91 on exercise number 18)
We can use a decimal point(.) to present the fraction. Just as places to the left of the decimal point represents units, tens, hundreds, and so on, those to the right of the decimal point represent places for tenths (1/10), hundredths (1/100), and so forth.
8/10=(0 x 1) + (8 x 1/10)=0.8
e. Attempting to Formalize Operation Methods (Idea of Algorithm)
Formal calculation requires one to have a solid understanding of methods, and the ability to mechanically perform calculations based on this understanding without having to think about the meaning of each stage, one after the other. This allows one to conserve cognitive effort, and to easily execute operations. This also applies to measurements and drawing figures. The mechanical execution of a predetermined set of procedures is referred to as using an “algorithm.” Thinking that attempts to create algorithms based on an understanding of procedures, is important.
(Mathematics for Junior High School 1 page 91 on exercise number 23)
Estimate the result of 88 x 41
Round up each number to the nearest multiple 10.
The nearest multiple 10 of 88 is 10 x 9.
The nearest multiple 10 of 41 is 10 x 4 = 40.
Hence, the best estimation of 88 x 41 is 90 x 40 = 3,600.
a. Attempting to Grasp the Big Picture of Objects and Operations, and Using the Result of this Understanding (Idea of Approximation)
A general understanding of results is effective for establishing a perspective on solving methods or on results, and for verifying results. By attaining a grasp on approximate numbers, amounts, or shapes, or doing approximate calculations or measurements, one can establish a perspective on results or methods, and verify results. This is the idea of approximation.
b. Focusing on Basic Rules and Properties (Idea of Fundamental Properties)
Calculation involves rules such as the commutative law, as well as a variety of properties such as “in division, the answer is not changed when one divides both the divided and dividing numbers by the same number.” Also, numbers have a variety of different properties such as multiples and divisors. Furthermore, figures and shapes have properties such as parallel and equal side lengths, area formulas, relationships between the units of amount, amount properties, proportional/inversely proportional amounts, and other numerous arithmetical or mathematical rules and properties. One must consider finding these, selecting the appropriate ones, and using them effectively. Thinking that focuses on these basic rules and properties, is therefore absolutely indispensable.
c. Attempting to Focus on What is Determined by One’s Decisions, Finding Rules of Relationships between Variables, and to Use the Same (Functional Thinking)
Meaning of Functional Thinking.
When one wants to know something about element y in set Y, or the characteristics and properties common to all elements of Y, in spite of the difficulty of clarifying this directly, one first thinks of object x, which is related to the elements in Y. By clarifying the relationship between x and y, functional thinking attempts to clarify these characteristics and properties.
d. Attempting to Express Propositions and Relationships as Formulas, and to Read their Meaning (Idea of Formulas)
· Idea of Expressing as Formulas
Thinking based on appreciation of expressing problems with the following types of formulas, which actively seeks to use these formulas, is important.
(Mathematics for Junior High School 1 page 91 on exercise number 299)
K = 2(p+l)
K is perimeter, p is length and l is width.
A rectangle ABCD in which AD=4 cm and CD = 6 cm
K=2(p+l)=2(6+4)=2 x 10=20
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